Suckle on the hope in light brassieres
The skinny love relationship is based on lust, where both lovers are bound together only by their sexual attraction to each other.
It is implied that the relationship is not built with sufficient emphasis on emotional, intellectual and spiritual connections between them.
Suckle is a verb, meaning: feed (a baby or young animal) from the breast or teat. with this being said, the speaker is referencing the hope that is felt when they have sex is what nourished the love, and the speaker might miss this, or not see why things had to end.
Vernon is anthropomorphizing skinny love in this line. She would have small breasts, requiring a light (small) brassiere (bra). Even a skinny mother gives some milk to her child. Vernon is using this imagery to say that he sees things in his current relationship, as incomplete as it is, that make him think it can still work and give him hope.
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